How do nuclei rotate? 1 2 3 symmetry axis The nucleus rotates as a whole. (collective degrees of...
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Transcript of How do nuclei rotate? 1 2 3 symmetry axis The nucleus rotates as a whole. (collective degrees of...
How do nuclei rotate?
1
2
3symmetry axis
The nucleus rotates as a whole. (collective degrees of freedom)
The nucleons move independentlyinside deformed potential (intrinsic degrees of freedom)
The nucleonic motion is much fasterthan the rotation (adiabatic approximation)
3
1
2
2i i
iJEenergy spheroid
Oblate and prolate quadrupole deformation
Choosing the vertical axis as the 3-axis one obtains the oblate by R1 = R2 > R3 and the prolate by R1 = R2 < R3 axially-symmetric quadrupole deformations
prolate deformation (β>0)
,1, *
0 YRR
,,sin
2
1,cos1, 2222200 YYYRR
00for
oblate deformation (β<0)
The Euler angles
It is important to recognize that for nuclei the intrinsic reference frame can have any orientation with respect to the lab reference frame as we can hardly control orientation of nuclei (although it is possible in some cases).
One way to specify the mutual orientation of two reference frames of the common origin is to use Euler angles.
(x, y, z) axes of lab frame(1,2,3) axes of intrinsic frame
The rotation from (x,y,z) to (x´,y´,z´) can be decomposed into three parts: a rotation by about the z axis to , a rotation of θ about the new y axis to , and finally a rotation of ψ about the new z axis .
zyx ,,
y zyx ,,
z
nspermutatio cyclic and ],[ :axes laboratory zyx JiJJ
nspermutatio cyclic and ],[ :axes fixedbody 321 JiJJ
zyxkJJ k ,, 0],[ 2
0],[ 3,2,1 0],[ 32 JJiJJ zi
)1( :numbers quantum 22 IIJMJ z
)1( :numbers quantum 223 IIJKJ
KMI ,,| :states
Quantization
KMI ,,| :seigenstate
function Wigner D iKIMK
iMIMK edeD )(),,(
),,(8
12,,|,, :rotor ofn orientatiofor
amplitudey probabilit2/1
2
I
MKDI
KMI
3
z
Rotational motion of a deformed nucleus
For a deformed nucleus with axial symmetry one obtains the same rotational frequency for the 1- and 2-axis. The Hamiltonian is given by
States with projections K and –K are degenerated
This fact is expressed in the nuclear wave function as a symmetric product
KI
KMKI
KIKMIMK DD
I
116
122/1
2
For K=0, only even-J are allowed, so that the wave function is given by a single term
00
2/1
28
12
IMIM D
I
3
23
1
23
23
1
2
2
ˆ
2
ˆˆ
2
ˆ
RRRRH
i i
irot
If the total angular momentum results only from the rotation (J = R), one obtains for the rotational energy of an axially symmetric nucleus by
12
2
IIErot
J
3
21
33.3)2(
)4(
1
12/4
E
ER
The nucleus does not have an orientation degree of freedom with respect to the symmetry axis
3
z
Note the characteristic, repeated patterns
Broad per spective on structural evolution:
neutron number
prot
on
num
ber
)2(
)4(
1
12/4
E
ER
γ-rays from a superdeformed band in 152Dy
Rotational motion of a deformed nucleus
axial symmetry:rotational axis symmetry axis
kinematic moment of inertia
E
I
EE
I
II
24242
22
1
22
12
KIIErot
dynamic moment of inertia
2
822
2
IEIE
rotational frequency
2
2
IEIE
dI
dE
3
z
Moment of inertia
z
R()
00
00
05.1900
53
4
R
R
R
RR
3/10 2.1 AR 0RR 201 Y 200 1 YRR
Rigid body moment of inertia:
dddrrrrR sin22
Irrotational flow moment of inertia:
22
8
9 oF RM
)32.01(5
2 2 oR RM
rigid
irrotational
Moment of inertia
Reduced transition probability
dMM abm
abm
ˆ*expectation value
ntrˆˆ im
mmm
abm MDM
QReZ
M m
16
12
4
3ˆ0
intr
wave function 0020 8
12
IMIM D
I
m
IMmmm
IM
fi
iiabmff dDMDD
IIMIMMI i
i
f
f
00intr
002ˆ
8
12120ˆ0
3321213321213
2
||12
83
33
2
22
1
11MIMMIIMIMMII
IdDDD I
MMI
MMI
MM
0intr00
ˆ0|00|12
120ˆ0
MIIMIMMMI
I
IMIMMI fiffifii
f
iii
abmff
Reduced transition probability
0intr00
ˆ0|00|12
120ˆ0
MIIMIMMMI
I
IMIMMI fiffifii
f
iii
abmff
Wigner-Eckart-Theorem (reduction of an expectation value):
0ˆ0
12
|0ˆ0 i
abf
f
ffifiiii
abmff IMI
I
MIMMMIMIMMI
00intr0 |ˆ0|001200 MIIIIEMI fiiif
special case: E2 transition I→I-2
4
3
122
1320,2 2
20ZeR
I
IIIEMI
reduced transition probability:
2020
12
1; if
ifi IEMI
IIIEB
2216
5
12122
132;2 Q
II
IIIIEB
Reduced transition probability
22513 ;210225.11
beIIEBMeVEs fi Transition probability:
2ln2/1 T
1521422 110161.8;2 sMeVEbeIIEB
iIETfi
half-life:
Weisskopf estimate:
223/461094.5;2 beAIIEB gsi
Hydrodynamical model
Reduced transition probability: 2
40
22
16
920;2
ReZ
EB 2
Excitation energy:2
62
2
E
Moment of inertia:2
8
9oF RM
2
2
3/1
2
20;2EA
ZconstEB
Hydrodynamical model
Reduced transition probability: 2
2
24
4222
220 33.097.036.01
4
320;2
ZeR
EB
Excitation energy: 24
22
20
2
2 3/53
8
RMAE
first indication of a hexadecapole deformation